Just like the actual puzzle workshop on Saturday, this Puzzle Presentation Post has two parts.
Unlike the actual workshop, I am doing the heavy lifting on both blog posts.
After my presentation was over we had a short break, did a little Ken Ken, and moved on to the second presenter, Joanne Becker. Joanne presented a fun powerpoint from a meeting she had attended. The questions were all based around the classic Knights & Knaves problem.
All you really need to know to do these problems is the following:
The country of Logica has two types of inhabitants:
• Knights always tell the truth
• Knaves always lie
We were given a variety of different scenarios and we had to determine who was a knight, who was a knave, and sometimes what replies the individual had given to our question. For example:
Alice and Bob live in Logica.
Alice says: At least one of us is a knave.
What are Alice and Bob?
We figured out the answers, and shared our methods with one another. It was fun. I really like these kinds of puzzles. On the car ride home Avery and I had a really interesting conversation about the differences between these types of logic puzzles and the kinds that I presented.
These puzzles are very “language dependent”. There is tricky wording–especially as the questions increase in complexity–that could present a barrier to students even after they have figured out the original premise (knights=truthtellers, knaves=liars). The puzzles I gave to the group have no language in them at all (unless you count numbers as words, which they aren’t; they are symbols). So, once you understand the premise for these puzzles the solving aspect is simply application of the rules. Whereas the Logica puzzles require a high level of interpretation to solve successfully.
Another thing Avery and I noted was that, as mathematically trained adults, neither of us had any problem with the fact that we didn’t know what the word “knave” meant. We understood that it wasn’t important to finding the answer and that we could get by with just knowing that they always lie. This is not always the case with students. A lot of students that I’ve had would be uncomfortable with the idea of working on a problem with unfamiliar terminology. They would want need to know what a knave was before they could even begin working on the problems. The terminology would prevent them from accessing the problem.
As people began presenting their problems another interesting pattern emerged. It was really hard to notate “knight” and “knave” quickly, so people changed the terminology. The knights became abbreviated as T (for truthteller) and the knaves as L (for liar). Again, this shows a comfort and familiarity with the idea that in math it doesn’t really matter that much what we call these individuals, so long as we can tell the difference between them. And that it’s okay to change the names. That this won’t change our ability to solve the problem–in fact it will help us solve the problem. Students might be hesitant to do this.
Finally, even with the changes in terminology it was still easy to get twisted up when talking about the problems, so participants developed ways to organize their thinking. Most of these involved some sort of chart or decision-tree type structure. I would love it if all of my students had the “make a chart” tool in their mental toolkit and felt comfortable whipping it out when they started working a problem like this. That would be so cool! I hate how students get trained to look to the problem for the “right” method for solving it (Not your students, I know.You all are legit.). Since there’s no chart in this problem, this type of student wouldn’t consider using a chart to help him/her answer this question.
None of this should be in any way construed as an argument that teachers should only use “language independent” puzzles with their students. One of the things I find really fascinating about teaching is how different modes of presentation affect how students approach a problem. And how I can make thoughtful decisions in how I introduce students to a new concept that will greatly aid–or not–their ability to access the material successfully. Elizabeth mentioned this in her comment on the last post. Access is key to the engagement issue. Students need to see a way in, an invitation to work on math. Once you’ve got them in the “mathematical door”, teaching them new things becomes much easier. But for a lot of students, this door has been slammed in their face on a regular basis. It is our job to gently pry their fingers off of the door frame and coax them to come inside and play. And to make sure that we do everything in our ability to not slam the door in their face again.